Theorem prsspwg 3859

Description: An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by Thierry Arnoux, 3-Oct-2016.) (Revised by NM, 18-Jan-2018.)

Assertion

Ref	Expression
prsspwg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶)))

Proof of Theorem prsspwg

Step	Hyp	Ref	Expression
1		prssg 3856	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ 𝒫 𝐶 ∧ 𝐵 ∈ 𝒫 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝒫 𝐶))
2		elpwg 3682	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐶 ↔ 𝐴 ⊆ 𝐶))
3		elpwg 3682	. . 3 ⊢ (𝐵 ∈ 𝑊 → (𝐵 ∈ 𝒫 𝐶 ↔ 𝐵 ⊆ 𝐶))
4	2, 3	bi2anan9 610	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ 𝒫 𝐶 ∧ 𝐵 ∈ 𝒫 𝐶) ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶)))
5	1, 4	bitr3d 190	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2205   ⊆ wss 3214  𝒫 cpw 3674  {cpr 3695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701

This theorem is referenced by: (None)